The main problem of protein folding is the "exponential search" (or multiple minimum) problem. To find the unique native structure (at the global minimum of free energy) requires a search over the whole conformational space, the size of which increases exponentially with chain length. But proteins fold much faster than this. How can a protein find the global optimum without a globally exhaustive search? We will address this question using a 2-dimensional short-chain HP (hydrophobic/polar) copolymer lattice model. This appears to be the simplest physical model of protein stability and structure that is amenable to rigorous study. It has been shown to mimic many aspects of protein behavior. Most importantly, the model has the same exponential search problem that real proteins have. We are exploring: (i) how the shape of a conformational space is determined by its amino acid sequence, (ii) the kinetic barriers to folding using Monte Carlo lattice kinetics and an analytical transition matrix approach, and (iii) distance measures that define structural similarities in real proteins and lattice models; these are necessary to define "nearness" to the native structure and "folding pathways." We have very exciting preliminary results suggesting that an answer to the search puzzle for the model may be near at hand. We find that some sequences fold faster than others, distinguished by characteristically shaped conformational spaces. We are finding some families of sequences that fold faster than exponentially and we have a preliminary algorithm that finds, for such sequences, the unique native lattice conformation and other deep minima without exhaustive search. We expect this model study to have significant implications for principles and algorithms of protein folding.